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\section{\usemenu{slac-pub-7144::context::slac-pub-7144-0-0-1}{Introduction}}\label{section::slac-pub-7144-0-0-1}
\label{sec:introd}
\setcounter{equation}{0}
In the Standard model (SM), flavor-changing neutral currents only
arise at the one-loop level.
This is why the corresponding rare B meson decays
are particularly sensitive to ``new physics''.
However, even
within the Standard model framework, one can use them
to constrain the Cabibbo-Kobayashi-Maskawa matrix
elements which involve the top-quark. For both these
reasons,
precise experimental and theoretical work on these
decays is required.
In 1993, $B \to K^* \gamma$ was the first rare B decay mode
measured
by the CLEO collaboration \cite{1}. Recently, also the first
measurement of
the inclusive photon energy
spectrum and the branching ratio
in the decay $\BGAMAXS$ was reported \cite{2}.
In contrast to the exclusive channels, the inclusive mode allows a less
model-dependent comparison with theory, because no specific
bound state model is needed for the final state. This opens
the road to a rigorous comparison with theory.
The data agrees with
the SM-based theoretical computations presented
in \cite{3,4,5},
given that there are large uncertainties in both the
experimental and the theoretical results.
In particular, the measured branching ratio $BR(B \to X_s \gamma)
= (2.32 \pm 0.67)
\times 10^{-4}$ \cite{2} overlaps with the
SM-based estimates in
\cite{3,4} and in \cite{6,7}.
In view of the expected increase in the experimental precision,
the calculations must be refined correspondingly in order to
allow quantitative statements about new
physics or standard model
parameters. So far, only the leading logarithmic
corrections have been worked out systematically.
In this paper we evaluate an important class of next
order corrections, which we will describe in detail below
\footnote{Some of the diagrams were calculated by Soares
\cite{8}}.
We start within the usual framework of an effective theory with
five quarks, obtained by integrating out the
heavier degrees of freedom which
in the standard model are the top quark and the $W$-boson.
The effective Hamiltonian includes
a complete set of dimension-6 operators relevant for the process
$b \to s \gamma$ (and $b \to s \g g$) \cite{9}
\begin{equation}
\label{heff}
H_{eff}(b \to s \gamma)
= - \frac{4 G_{F}}{\sqrt{2}} \, \lambda_{t} \, \sum_{j=1}^{8}
C_{j}(\mu) \, O_j(\mu) \quad ,
\end{equation}
with
$G_F$ being the Fermi
coupling constant and
$C_{j}(\mu) $ being the Wilson coefficients evaluated at the scale $\mu$,
and $\lambda_t=V_{tb}V_{ts}^*$ with $V_{ij}$ being the
CKM matrix elements.
The operators $O_j$ are as follows:
\bea
\label{operators}
O_1 &=& \left( \bar{c}_{L \b} \g^\m b_{L \a} \right) \,
\left( \bar{s}_{L \a} \g_\m c_{L \b} \right)\,, \nonumber \\
O_2 &=& \left( \bar{c}_{L \a} \g^\m b_{L \a} \right) \,
\left( \bar{s}_{L \b} \g_\m c_{L \b} \right) \,,\nonumber \\
O_3 &=& \left( \bar{s}_{L \a} \g^\m b_{L \a} \right) \, \left[
\left( \bar{u}_{L \b} \g_\m u_{L \b} \right) + ... +
\left( \bar{b}_{L \b} \g_\m b_{L \b} \right) \right] \,,
\nonumber \\
O_4 &=& \left( \bar{s}_{L \a} \g^\m b_{L \b} \right) \, \left[
\left( \bar{u}_{L \b} \g_\m u_{L \a} \right) + ... +
\left( \bar{b}_{L \b} \g_\m b_{L \a} \right) \right] \,,
\nonumber \\
O_5 &=& \left( \bar{s}_{L \a} \g^\m b_{L \a} \right) \, \left[
\left( \bar{u}_{R \b} \g_\m u_{R \b} \right) + ... +
\left( \bar{b}_{R \b} \g_\m b_{R \b} \right) \right] \,,
\nonumber \\
O_6 &=& \left( \bar{s}_{L \a} \g^\m b_{L \b} \right) \, \left[
\left( \bar{u}_{R \b} \g_\m u_{R \a} \right) + ... +
\left( \bar{b}_{R \b} \g_\m b_{R \a} \right) \right] \,,
\nonumber \\
O_7 &=& (e/16\p^{2}) \, \bar{s}_{\a} \, \sigma^{\m \n}
\, (m_{b}(\mu) R + m_{s}(\mu) L) \, b_{\a} \ F_{\m \n} \,,
\nonumber \\
O_8 &=& (g_s/16\p^{2}) \, \bar{s}_{\a} \, \sigma^{\m \n}
\, (m_{b}(\mu) R + m_{s}(\mu) L) \, (\l^A_{\a \b}/2) \,b_{\b}
\ G^A_{\m \n} \quad .
\nonumber \\
\eea
In the dipole type
operators $O_7$ and $O_8$,
$e$ and $F_{\m \n}$ ($g_s$ and $G^A_{\m \n}$)
denote the electromagnetic (strong)
coupling constant and
field strength
tensor, respectively. $L=(1-\g_5)/2$ and $R=(1+\g_5)/2$
stand for the left and right-handed projection operators.
It should be stressed in this context that the explicit
mass factors in $O_7$
and $O_8$ are the running quark masses.
QCD corrections to the decay rate for $b \to s \g$
bring in
large logarithms of the form $\a_s^n(m_W) \, \log^m(m_b/M)$,
where $M=m_t$ or $m_W$ and $m \le n$ (with $n=0,1,2,...$).
One can systematically resum these large terms by renormalization
group techniques. Usually, one matches the full standard model theory
with the effective theory at the scale $m_W$. At this scale,
the large logarithms generated by matrix elements in the
effective theory are the same ones
as in the full theory. Consequently, the
Wilson coefficients only contain small QCD corrections.
Using the renormalization group equation, the Wilson coefficients
are then calculated at the scale $\mu \approx m_b$,
the relevant scale for a $B$ meson decay.
At this scale
the large logarithms are contained in
the Wilson coefficients
while the matrix elements of the operators are free
of them.
As noted, so far the decay rate for $b \to s \gamma$ has been
systematically
calculated only to leading logarithmic accuracy i.e., $m=n$.
To this precision it is consistent to perform the 'matching' of the
effective and full theory
without taking into account QCD-corrections
\cite{10}
and to calculate
the anomalous dimension matrix to order $\a_s$ \cite{11}.
The corresponding leading logarithmic Wilson coefficients
are given explicitly in
\cite{6,12}.
Their numerical values in the naive dimensional scheme (NDR) are
listed in table 1 for different values of the renormalization
scale $\mu$. The leading logarithmic contribution to the
decay matrix element is then obtained by calculating the
tree-level matrix element of the operator $C_7 O_7$ and
the one-loop matrix elements of the four-Fermi operators
$C_i O_i$ ($i=1,...,6$).
In the NDR scheme
the latter
can be absorbed into a redefinition of $C_7 \to C_7^{eff}$
\footnote{For the analogous $b \to s g$ transition, the effects
of the four-Fermi operators can be absorbed by the shift
$C_8 \to C_8^{eff}=C_8 + C_5$.}
\be
\label{C78eff}
C_7^{eff} \equiv C_7 + Q_d \, C_5 + 3 Q_d \, C_6 \quad .
\ee
In the `t Hooft-Veltman scheme (HV) \cite{13},
the contribution of the four-Fermi
operators vanishes. The Wilson coefficients
$C_7$ and $C_8$ in the HV scheme
are identical to $C_7^{eff}$ and $C_8^{eff}$ in the
NDR scheme.
Consequently, the complete leading logarithmic result
for the decay amplitude $b \to s \g$ is
indeed scheme independent.
Since the first order calculations have
large
scale uncertainties,
it is important to take into account the next-to-leading
order corrections. They are most prominent in the
photon energy spectrum. While it is a
$\delta$-function
(which is smeared out by the Fermi motion of the $b$-quark inside
the $B$ meson) in the leading order, Bremsstrahlung corrections, i.e.
the process $b \to s \gamma g$, broaden the shape of the spectrum
substantially. Therefore, these
important corrections
have been taken into account
for the contributions of the operators $O_7$ and $O_2$ some time
ago \cite{3} and recently
also of the full
operator basis \cite{4,14,15}.
As expected,
the contributions of $O_7$
and $O_2$ are by far the most important ones, especially
in the experimentally accessible part of the spectrum.
Also those
(next-to-leading) corrections,
which are necessary to cancel the
infrared
(and collinear) singularities of
the Bremsstrahlung diagrams were
included.
These are the virtual gluon corrections to the contribution
of the operator $O_7$ for $b \to s \g$ and the virtual
photon corrections to $O_8$ for $b \to s g$.
A complete next-to-leading calculation implies
two classes of improvements: First,
the Wilson coefficients to next-leading
order at the scale $\mu \approx m_b$ are required. To this end
the matching
with the full theory (at $\mu=m_W$)
must be done at the $O(\a_s)$ level and the renormalization
group equation has to be solved using the anomalous dimension
matrix calculated up to order $\a_s^2$.
Second,
the virtual $O(\a_s)$ corrections for the matrix element (at scale
$\mu \approx m_b$)
must be evaluated and combined with the Bremsstrahlung corrections.
The higher order matching has been calculated in ref. \cite{16}
and work on the Wilson coefficients is in progress.
In this paper we will evaluate all the virtual
correction beyond those evaluated already in connection with the
Bremsstrahlung process. We expect them
to reduce substantially
the strong scale dependence of the leading order calculation.
Among the four-Fermi operators only $O_2$
contributes sizeably and we calculate only
its virtual corrections
to the matrix element for $b \to s \g$.
The matrix element $O_1$ vanishes
because of color,
\begin{table}[htb]
\label{coeff}
\begin{center}
\begin{tabular}{| c | c | r | r | r | }
\hline
$C_i(\mu)$ & $\mu=m_W$ & $\mu=10.0$ GeV
& $\mu=5.0$ GeV
& $\mu=2.5$ GeV\\
\hline \hline
$C_1$ & $0.0$ & $-0.149$ & $-0.218$ & $-0.305$ \\
$C_2$ & $1.0$ & $1.059$ & $1.092$ & $1.138$ \\
$C_3$ & $0.0$ & $0.006$ & $0.010$ & $0.014$ \\
$C_4$ & $0.0$ & $-0.016$ & $-0.023$ & $-0.031$ \\
$C_5$ & $0.0$ & $0.005$ & $0.007$ & $0.009$ \\
$C_6$ & $0.0$ & $-0.018$ & $-0.027$ & $-0.040$ \\
$C_7$ & $-0.192$ & $-0.285$ & $-0.324$ & $-0.371$ \\
$C_8$ & $-0.096$ & $-0.136$ & $-0.150$ & $-0.166$ \\
$C_7^{eff}$ & $-0.192$ & $-0.268$ & $-0.299$ & $-0.334$ \\
$C_8^{eff}$ & $-0.096$ & $-0.131$ & $-0.143$ & $-0.157$ \\
\hline
\end{tabular}
\end{center}
\caption{Leading logarithmic Wilson coefficients $C_i(\mu )$
at the matching scale $\mu=m_W=80.33$ GeV
and at three other scales, $\mu = 10.0$ GeV,
$\mu =5.0$ GeV and $\mu = 2.5$ GeV. For $\a_s(\mu)$
(in the $\overline{MS}$ scheme) we used the
one-loop expression with 5 flavors and $\a_s(m_Z)=0.117$.
The entries correspond to the top
quark mass
$\overline{m_t}(m_{t,pole})=170$ GeV (equivalent to
$m_{t,pole}= 180 $ GeV).}
\label{table1}
\end{table}
and the
penguin induced four-Fermi operators $O_3,...., O_6$ can be
neglected \footnote{
This omission will be a source of a slight scheme and scale dependence
of the next-to-leading order result.}
because their Wilson
coefficients
\footnote{It is consistent to calculate the corrections
using the leading logarithmic Wilson coefficients.}
are much smaller than $C_2$, as illustrated
in table \docLink{slac-pub-7144-0-0-1.tcx}[coeff]{1}.
However, we do take into account the virtual $O(\a_s)$
corrections to $b \to s \g$ associated with the
magnetic operators $O_7$ (which has already been calculated in the
literature) and $O_8$ (which is new).
Since the corrections to $O_7$ and $O_8$ are one-loop diagrams,
they are relatively easy to work out.
In contrast, the corrections to $O_2$, involve two-loop
diagrams, since this operator itself only contributes
at the one-loop level.
Since the virtual and Bremsstrahlung
corrections to the matrix elements are only one (well-defined)
part of the whole next-to-leading program,
we expect that this contribution alone will depend on
the renormalization scheme used. Even within the modified minimal
subtraction scheme $(\overline{MS})$ used here,
we expect that two different
``prescriptions'' how to treat $\gamma_5$, will lead to different
answers. Since previous calculations of the Bremsstrahlung
diagrams have been done in the NDR scheme and also the leading
logarithmic Wilson coefficients are available in this scheme,
we also use it here.
For future checks, however, we also consider in Appendix A the
corresponding calculation in the HV scheme.
The remainder of this paper
is organized as follows. In section 2 we give the
two-loop corrections for $b \to s \g$ based on the operator $O_2$
together with the counterterm contributions.
In section 3 the virtual corrections for $b \to s \g$
based on $O_7$ are reviewed including
some of the Bremsstrahlung corrections.
Then, in section 4 we calculate the one-loop
corrections to $b \to s \g$
associated with $O_8$. Section 5 contains the results
for the branching ratio for $b \to s \g (g)$ and especially the
drastic reduction of the renormalization scale dependence
due to the new contributions.
Appendix A contains the result of the
$O_2$ two-loop calculation in the HV scheme and, finally,
to make the paper self-contained, we include in Appendix B
the Bremsstrahlung corrections to the operators $O_2$, $O_7$
and $O_8$.
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